Euclidean Distances Between Points on a Sphere

Question

Let $r$, $g$, and $b$ be positive real numbers such that $r \geq g \geq b$. Consider a sphere of radius $r$ centered at the origin. If M is any point on the sphere, prove that there exist two points P and Q on the sphere such that MP = MQ = g and PQ = b.

Answer 1

You can use the spherical law of cosines to compute the angles of the spherical triangle $MPQ$ from the angles subtended by each of the edges, which in turn you can compute from the lengths you are given. So unless one of these steps becomes impossible (e.g. some cosine would have to have an absolute value greater than one), you know that suitable points will exist.

Answer 2

You can draw a small circle centered at $M$ with all the points at distance $g$. You know it exists because as the circle moves away from $M$ the distance increases from $0\lt g$ to $2r\gt g$. Now call some point on the small circle $P$. Let $Q$ move away from $P$ along the circle until the distance is $b$ and you are done. You have to show that the diameter of the small circle is greater than $b$, which is easiest if you show it is greater than $g$. Draw a plane through $M$ perpendicular to the plane of the small circle and show the angle at $M$ is greater than $\frac \pi 3$